Accessing the Gluon Momentum Fraction of Nucleons through the Gradient Flow

Abstract

We calculate the gluon momentum fraction of the nucleon using lattice quantum chromodynamics (QCD), with a nonperturbative renormalization technique based on the gradient flow. The gluon momentum fraction is determined on a single Wilson-clover ensemble using Nf = 2+1 flavors with pion mass 358 MeV and lattice spacing 0.094 fm. We employ the variational method to reduce excited-state contamination and apply the distillation framework to ensure a large operator basis. To reduce systematic uncertainties, we apply Bayesian model averaging to all fit procedures. We apply matching coefficients to the flow-time dependent lattice results to recover the gluon momentum fraction in the MS-scheme at 2 GeV. Our final result is <x>_g(μ= 2 GeV) = 0.482(35), where we quote only statistical uncertainties.

Figures (10)

• Figure 1: Blue data points: lattice determinations of the gluon momentum fraction referenced in Table \ref{['table:glu-mom-frac_latt_comparison']}, including the result presented in this work in red. Black data points: determinations of the gluon momentum fraction calculated using PDF data from LHAPDF Buckley:2014ana, and from recent JAM results Anderson:2024evk. The grey vertical band represents the unweighted mean of these results, $\langle x\rangle_g^{\mathrm{exp.}}(\mu=2\,\mathrm{GeV}) = 0.409(7)$, to illustrate the target precision for future lattice calculations.
• Figure 2: Two-point correlation functions under distillation. The perambulators, red lines, are factored from the elementals, the blue and green ovals. Different interpolators can be easily swapped in and out for the source/sink operators once the expensive perambulators have been computed. The gluon operator, constructed from the gauge field strength tensor, is shown as a set of four black squares.
• Figure 3: Analysis of the dispersion relation for the nucleon ground state. The data displayed in the main plot are defined as $\delta E \equiv \left(E_0(\vec{p}) / \sqrt{E_0(0)^2 + \vec{p}^2} - 1\right) \times 10^3$. The band around unity is set to match the uncertainty of the zero-momentum data. The inset shows the energy levels in lattice units and the analytic dispersion relation $E_0(\vec{p}) = \sqrt{E_0(0)^2 + \vec{p}^{\,2}}$, shown in black. The energies are $aE_0 \in \left[0.53281(61), 0.56781(72), 0.6614(11)\right]$.
• Figure 4: Principal correlators for the ground-state nucleon at rest, divided by the leading exponential. Deviations from unity characterize excited-state contaminations.
• Figure 5: Ratios of the rotated and projected three- to two-point correlation matrices for three different flow-times, $\tau / a^2 \in \left\{ 1.2, 1.6, 1.8\right\}$.